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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 20. Выпуск 2.
УДК 514.76 DOI 10.22405/2226-8383-2019-20-2-7-21
Обобщенные многообразия Кенмоцу постоянного типа
Абу-Салеем Ахмад, А. Р. Рустанов, Т. Л. Мелехина
Абу-Салеем Ахмад — доцент кафедры математики (геометрия и топология), Университет Аль-Байт (Мафрак, Иордания). e-mail: [email protected]
Рустанов Алигаджи Рабаданович — кандидат физико-математических наук, старший преподаватель кафедры прикладной математики, Национальный исследовательский Московский государственный строительный университет (НИУ МГСУ), Институт фундаментального образования (ИФО МГСУ) (г. Москва). e-mail: [email protected]
Мелехина Татьяна Леонидовна — кандидат физико-математических наук, доцент кафедры анализа данных, принятия решений и финансовых технологий, Финансовый университет при Правительстве Российской Федерации (г. Москва). e-mail: [email protected]
Аннотация
В работе мы рассматриваем обобщенные многообразия Кенмоцу, мы вводим четвертое и пятое фундаментальные тождества обобщенных многообразий Кенмоцу, вводятся первый и второй структурные тензоры обобщенных многообразий Кенмоцу и доказаны их свойства, вводится понятие присоединенной Q-алгебры для обобщенных многообразий Кенмоцу. Доказано, что обобщенное многообразие Кенмоцу, а также специальные обобщенные многообразия Кенмоцу II рода имеют антикоммутативную присоединенную Q-алгебру. А многообразия Кенмоцу и специальные обобщенные многообразия Кенмоцу I рода имеют абелеву присоединенную Q-алгебру. Вводится контактный аналог постоянства типа и подробно исследуются обобщенные многообразия Кенмоцу постоянного типа. Получены условия точечного постоянства типа обобщенных многообразий Кенмоцу на пространстве присоединенной G-структуры. Доказано, что класс GK-многообразий нулевого постоянного типа совпадает с классом многообразий Кенмоцу, а класс GK-многообразий ненулевого постоянного типа конциркулярным преобразованием переводит-ся в почти контактное метрическое многообразие локально эквивалентное произведению шестимерного собственного NK-многообразия на вещественную прямую.
Ключевые слова: многообразия Кенмоцу, обобщенные многообразия Кенмоцу, специальные обобщенные многообразия Кенмоцу I рода, специальные обобщенные многообразия Кенмоцу II рода, GK-многообразия постоянного типа, точнейше косимплектическое многообразие.
Библиография: 15 названий. Для цитирования:
Абу-Салеем Ахмад, А. Р. Рустанов, Т. Л. Мелехина. Обобщенные многообразия Кенмоцу постоянного типа // Чебышевский сборник, 2019, т. 20, вып. 2, с. 7-21.
CHEBYSHEVSKII SBORNIK Vol. 20. No. 2.
UDC 514.76 DOI 10.22405/2226-8383-2019-20-2-7-21
Generalized Kenmotsu manifold constancy of type
Abu-Saleem Ahmad, A. R. Rustanov, T. L. Melekhina
Abu-Saleem Ahmad — Professor, Department of Mathematics , Al al-Bavt University (Mafraq, Jordan).
e-mail: [email protected]
Rustanov Aligadzhi Rabadanovich — Candidate of Physics and Mathematics, Associate Professor, Department of Applied Mathematics, Federal state budget educational institution of higher education "NATIONAL RESEARCH MOSCOW STATE UNIVERSITY OF CIVIL ENGINEERING"(NRU MGSU) (Moscow). e-mail: [email protected]
Melekhina Tatyana Leonidovna — Candidate of Physics and Mathematics, Associate Professor of Data Analysis, Decision Making and Financial Technologies, Financial University under the Government of the Russian Federation (Moscow). e-mail: [email protected]
Abstract
In this work we consider generalized Kenmotsu manifolds, we introduce: the fourth and the fifth fundamental identities of generalized Kenmotsu manifolds; the first and the second structural tensors of generalized Kenmotsu manifolds (and we prove their properties); the concept of adjoint Q-algebra for generalized Kenmotsu manifolds. We prove that generalized Kenmotsu manifolds and the II kind special generalized Kenmotsu manifolds have anticommutative adjoint Q-algebra. And the Kenmotsu manifolds and the I kind special generalized Kenmotsu manifolds have Abelian adjoint Q-algebra. The type constancy contact analog is introduced and the constant-type generalized Kenmotsu manifolds are thoroughly-examined. We have identified the type point constancy conditions of the generalized Kenmotsu manifolds in the adjoint G-structure space. We prove that the zero constant type GK-manifold class coincides with the Kenmotsu manifold class and the non-zero constant type GK-manifold class can be concircularly transformed into the almost contact metric manifolds locally equivalent to the product of the six dimensional NK-eigenmanifold and the real straight line.
Keywords: Kenmotsu manifolds, generalized Kenmotsu manifolds, the I kind special generalized Kenmotsu manifolds, the II kind special generalized Kenmotsu manifolds, constant type GK-manifolds, most precise cosymplectic manifold.
Bibliography: 15 titles. For citation:
Abu-Saleem Ahmad, A. R. Rustanov, T. L. Melekhina, 2019, "Generalized Kenmotsu manifold constancy of type" , Chebyshevskii sbornik, vol. 20, no. 2, pp. 7-21.
1. Introduction
Contact and almost contact structures are one of the most substantial examples of differential geometrical structures. Nevertheless the most important geometrical properties of almost contact
metric manifolds are revealed when the additional limitations are applied to them. The most natural limitation is the isotropv condition.
The almost Hermitian manifold isotropv can be characterized by the constancy of their type fl], [2]. The type constancy of approximately Keller manifolds was introduced by A. Gray [2] and proved to be very useful for approximately Keller manifold studying. The complete characteristics of approximately Keller manifolds were obtained by V.F. Kirichenko [3].
In this work we consider the type constancy contact analog for generalized Kenmotsu manifolds which were introduced in the thesis work of Umnova S.V. [4]. In the work [4] Umnova S.V. singles out two subsets of generalized Kenmotsu manifolds, called special generalized Kenmotsu manifolds (shorter, SGK-) of the I and II kind. In the work [4] it's proved that generalized Kenmotsu manifolds of the constant curvature are the Kenmotsu manifolds [5] of the constant curvature (-1). Moreover, it's proved that the class of SGK- manifolds of the II kind coincides with the class of almost contact metrical manifolds received from the most precise cosvmplectic manifolds [6] through canonical transformation of the most precise cosvmplectic structure and the local construction of these manifolds is given. In this article we explore the generalized Kenmotsu manifolds of the constant type and give their complete local characteristics.
This article is organized in the next way. In Section 2 we present the preliminaries for the next statements, build the space of adjoint G-structure and put down the first structural equation group on the adjoint G-structure space. In Section 3 we give the definition of generalized Kenmotsu manifolds, give the full structural equation group, we prove that a generalized Kenmotsu manifold in a dimension different from 5 is a special generalized Kenmotsu manifold of the II kind and we provide fundamental identities of generalized Kenmotsu manifolds.
In the Section 4 we consider the adjoint Q-algebra of a generalized Kenmotsu manifold. We establish the theorem which is the basic result of the present paragraph and which means that the adjoint Q-algebra of a generalized Kenmotsu manifold is anticommutative. Three Conclusions are given for this theorem; they characterize the adjoint Q-algebras of generalized Kenmotsu manifold special cases.
In the Section 5 we explore the generalized Kenmotsu manifolds of a constant type. It's proved that the generalized Kenmotsu manifolds of the non-zero constant type are the generalized Kenmotsu manifolds of the II kind and we received their local structure. Generalized Kenmotsu manifolds of the non-zero type coincide with the Kenmotsu manifolds.
2. Preliminaries
Assume, that M is a smooth manifold of dimension 2n + 1 X(M) - C^ is a module of smooth vector fields on the manifold M. Further all manifolds, tensor fields and the like are supposed to be smooth of the class C(X.
Definition 1. ¡7] An almost contact structure on the manifold M is the trip let ("q, $) of tensor fields on this manifold where "q is a differential 1-form which is called a contact form of structure, £ is a vector field which is called characteristical, $ is the endomorphism of module X(M) which is called structural endomorphism. Here
1) ^(0 = 1; 2) V o $ = 0; 3) $(0=0; 4) $1 = -id + V <g> (1)
Besides, if such a Riemannian structure g = (■, •) is fixed on M that
{$X, $Y) = (X,Y)- r](X )rj(Y), X,Y eX (M). (2)
quadruple $g = (■, ■)) is called almost contact metrical (shorter, AC-) structure. The manifold where an almost contact (metrical) structure is fixed is called an almost contact (metrical (shorter, AC-)) manifold.
Skew-symmetric tensor Q(X,Y) = {X, $Y} ,X,Y e X(M) is called the fundamental form of an AC-structure [7].
Assume, that $ g = {■, ■}) is an almost contact metrical structure on the manifold M2n+1. In the module ^(M) two mutually complementing projections m = ® £ and I = id — m = —$2 [8]; are internally defined in the way that #(M)=C ©M, where C = Im($) = kerq - a so-called contact distribution, dimC=2n, M = I mm = ker($) = L(£) - a linear span of a structure vector (also I and m are the projectors for sub modules C, M accordingly). Obviously that distributions C and M are invariant towards $ and are mutually orthogonal. It's also obvious that $2 = —id, <1X, <1Y^ = {X,Y}, X,Ye ^(M), where <1 = $lc. Consequently {&P,gPlc} is a Hermitian structure on the space Cp (p - some point of M).
The complexification of the module A'(M) disintegrates into a direct sum
X(Mf = Di- © © D%
of structural endomorphism $ own spaces corresponding to their own values — 1 and 0
accordingly. Besides, the projectors for the summands of this direct sum will be endomorphisms ([7], [8]): -k = a o I = — T($2 + V—1$), tt = a o I = T(—$2 + V—1$), m = id + $2, where a = 2(id — V-1$), t = 22id + ^—1$).
The depictions ap : Cp ^ D^-1 and ap : Cp ^ D-^-1 are the isomorphism and the anti-isomorphism of Hermitian spaces, accordingly. That's why to every point p e M2n+1 it's possible to add a family of reference frames of space TP(M )c of the kind (p, ,t1,..., en, ei,..., e^) where ea = V2ap(ea), e^ = V2ap(ea), ao = (p where {ea} is the orthonormalized base of the Hermitian space Cp. This reference frame is called an A-reference frame [8]. It's clear that matrices are the element of tensors $p and gp in the A-reference frame; they have the according forms of:
0 0 0 \ /10 0
($)) = I 0 V—lln 0 I , (gl3) = I 0 0 In I , (3)
0 0 —V—Hn J V 0 in 0
where In is the unity matrix of the n-order. It is well known [7], [8] that the aggregate of such reference frames determines the G-structure on M with structural group {1} x U (n) represented bv
I 100 I
such matrices as 0 A 0 , where A e U (n). This G-structure is called adjoint [7], [8].
V 0 0 A )
Assume that (M2n+1, >q,£, $ g = {■, ■}) is an almost contact metrical manifold. We will make a convention that in this entire work indices i, j, k, I run through values from 1 to 2n, indices a, b, c, d, - through values from 1 to n, and let's assume that a = a + n, a = a, 0 = 0, unless it is stated otherwise. Let (U, <p) be a local chart on manifold M. According to the tensor analysis fundamental theorem the assignment of the structural endomorphism $ and the Riemann structure g = {■, ■} on the manifold M inspires on the total BM space of a bundle of frames above M a system of functions {$lj}, {gij}, complying in the coordinate neighborhood W = -k-1(U) C BM with a differential equation system of the following form
d$) + $)— $1 dk3 = ${j, k)luk, dgij — gkjd% — gikdk3 = gl3,kuk, (4)
where {us1}, {d%-} are the components of the solder forms and the Riemannian connection, correspondingly; $%jk, gij,k are the components of the covariant tens or differential $ and g in this connection correspondingly. Moreover, because of the Riemannian connection definition Vg = 0, and it means that
9ij,k =
(5)
With account of (3) and (5) relations (4) on the adjoint G-structure space can be put down as
[7], f8]
^ = 0, ^k = 0, $0,k = 0, °t = %,= - ,
6% = V—l^kujk, 0g = , (6)
% = —V—t^Uuk, el = ,
9\ + 9i = 0, 90 = o.
3 3 1
Besides, note that because of the real type nature of the corresponding forms and tensors
(xil = u}\ 93 = 9X-, k = , where t ^ t is a complex conjugation operator.
3 3 3} 3 ,k
WTith regard to the obtained relations the first structural equation group of the Riemannian connection dwl = —93 A us3 on the adjoint G-structure space of the almost contact metrical manifold can be formulated as the first group of almost contact metrical manifold structural equations [7],
[8]:
du = Cabwa A œb + Cabwa A wb + Cbawa A wb + Cau A ua + Cau A wa;
Qa yb
dwa = —9% A wb + Babcwc A wb + Babcwb A wc + Babb + Babu A wb; (7)
dWa = 9ba A Wb + BahA Wb + Babc^ A Ulc + BabU A Wb + BabU A Ub,
where w = w0 = ft is a natural projection of the adjoint G-structure space on the manifold
M, Ui = gi
Babc = — $a, ^a6c = $a*, = .,
Babe = — ^ $a,C ^ a6 = V—l$a>6, Bgh = l^^a^,
= v——1Bab = —V—i($a,6—1 $a,o), (8)
\aA]
Cab = Cab = -^^P ^ = -V-!^? + *°j)
c- = , Ca = v-i*°.
3. Generalized Kenmotsu manifolds
Assume that (M2n+l, $, (,ri,g = (-, ■}) is an almost contact metrical manifold. Definition 2. [4]. The class of almost contact metrical manifolds characterized by equality
Vx+ Vy($)X = —r](Y— r](X)$Y; X,Y eX(M), (9)
is called generalized Kenmotsu manifolds (shorter, GK- manifolds).
The full group of GK-manifold structural equations takes the form [9]:
1) du = FabWa A ub + FabUa A ub;
2) dua = -91 A ub + Cabcub A uc - 3Fabu A ub + ôfa A ub;
3) dUa = 9ba A Ub + CabcUb A Wc - f FabU A Ub + ^W A Ub;
4) d9l = -9ac A 9% + {Aa£ - 2CadhChbc - f FadFbc)wc A ud + +(-1ôabFcd + f5acFdb + 3S",Fbc)wc A ud + ( 15abFcd - 35cbFda - 35dFac)uc A Ud;
5) dCabc + CdbcQad + Cadcdbd + Cabd9cd = Cabcdud - 2ó[dFbc]ud - Cabcu; (10)
6) dCabc - Cdhc^a - ^adc^b - Cabd^ = Cabcd^d - 2àdaFbc]ud - Cabc^',
7) dFab + Fcbea + Fac0b = -2Fabw;
8) dFab - Fcbdca - Fac0°b = -2FabU]
where
Cabc = ^ Cabe = - ^ ñ,c ; C[a6c] = Cabc; C[abc] = Cabe; C^ = Cabc; Fab = Fab = -V^^b ;
Fab + Fba = 0; Fab + Fba = 0; ~Fa = Fab; Aadr] = A^ = 0; C a[bcd] = 3 Fa\bFcd]. Ca[bcd] = § ^^ FadCdbc = FadCdbc = 0.
Proposition 1. [9]. IfCabc = Cabc = 0 md Fab = Fab = 0, then a GK-manifold is a Kenmotsu manifold.
Differentiating externally (IO4 - IOô) we get:
i) dAaf+Affla+Aa¿od - Aidah - ¿asíeh = Aadch++^
2) dCab°d + ÇJhbcdga + qahedgb + qabhdQc + qabchgd _
hhhh = c a bcdh^h + C abcdO^;
3) dCabcd - ChbcdOa - CahcdOb - Cabhd^ - Cabch^'d =
= Cabcdh^h + CabcdO^.
Herewith:
^ ¿au]=0; 2) Aalfh]=0;
3) = -2Afc + FadFbc - 2^FdhFhc - 26'FdhFhb - 2ódFahFhc; 4) (Aca¡c - 2Cagf Cfb\c)C\g\dh] = 0; 5) (A^ - 2Ca^f\Cfbg)Clgldh] = 0;
6) (Aa[c - §FahFb[c)Fm = 0; 7) (A'^d - |FadFbh)Flhlc] = 0;
3
8) çabc[dh] _ _2{(jabcpdh + 1 (çadhpbc + çbdhpca + çcdhpab +
abhpdc + çi aedp hb + çi ahc pdb + çi dbc pah + çi hbcpda)}; g) q abcdO _ _(2C abed + pab ped + pac pdb + pad pbc);
10) CabcgCgdh = 0; ii) CabchFhd = 0; (12)
12) Cabc[dh] = -2{CabcFdh + 1 (CadhFbc + CbdhFca + CcdhFab + CdbcFah + +ChbcFda + CacdFhb + Cahc Fdb + CabdFch + CabhFdc )}; 13) Cabcdh = (A^h - 2Cghf Cfad)Cgbc + (Ag* - 2CghfCfbd)Cagc +
+(Agdh - 2Cghf CfCd)Cabg;
144) CabcdO = -(2Cabcd + Fab Fcd + Fac Fdb + FadFbc ^ 15) CabcgCgdh = 0; 16) CabchFhd = 0.
Differentiating externally (IO7) and (10s), we get:
l) Aadphb _ ^bdpha _ 3padp^ phb _ 3pbdp^ pha;
2) 2FabFcd _ (^ch — 5aFdh)Fhb + (6bFdh — 5bdFch)Fha;
3) 2pabpcd _ pacpdb + padpbc. ^gj
4) FahAhd — FbhAhd _ 2 (FadFbh — FbdFah)Fhc; 5) 2FabFcd _ (5cbFdh — 5^Fch)Fha + (5iFch — ^Fdh)Fhb; 6) 2FabFed _ FacFdb + FadFbc.
As a useful consequence (132) we prove the next theorem.
Theorem 1. [9] A GK-manifold of dimension different from 5 is a SGK-manifold of the II kind.
The identity
(Aabfc — 2CafCfb[c)CMdh] _ 0 (14)
we call the first fundamental equality of GK-manifolds [10].
The identity
(Aah — 3FahFb[c)F{hd _ 0 (15)
we call the second fundamental equality of GK-manifolds [10].
The identity
2pab pc d _ pac pdb + padpbc (16)
we call the third fundamental equality of GK-manifolds [10].
The identity
FadCdbc _ FadCdbc _ 0 (17)
we call the forth fundamental equality of GK-manifolds.
The identity
CabcgC^dh _ 0 (18)
we call the fifth fundamental equality of GK-manifolds.
The system of functions (Cabc,Cabc) determines the tensor (2,1) which is called the first structural tensor, the system of functions (Fab, Fab) determines the tensor (1,0) which is called the second structural tensor. The structural tensors of a GK-structure have the following equations [10]:
l) C(X, Y) _ — 1 $ o _ — 2o ($)$2X; (19)
2) F(X) _ $ o V$2X($)£ — $2X _ —$ o Vx($)£ — $2X _ —Vx{ — $2X _ _ —$2 o V$x ($)C — $2X _ —$ o V$x ($)C — $2X; X,Y e X (M).
Definition 3. [4] A generalized Kenmotsu manifold with a zero first structural tensor is called a special generalized Kenmotsu manifold (SGK-, for short) of the I kind.
Definition 4. [4] A generalized Kenmotsu manifold with a zero second structural tensor is called a special generalized Kenmotsu manifold (SGK-, for short) of the II kind.
4. Q-algebras of generalized Kenmotsu manifolds
In this section we discuss the Q-algebra adjoint to a GK-manifold.
Definition 5. [11] A Q-algebra is a triplet {V, ((■, ■}}, *} where V is a module of the commutative associative ring K with nontrivial involution; ((■, ■}} is a non-degenerated Hermitian form on V; * is a binary operation * : V x V ^ V, antilinear for each argument for which the Q-algebra axiom is accomplished ((X * Y, Z}} + ((Z * X, Y}} = 0, X,Y,Z £ V.
If K=C, then Q-algebra V is called complex.
Definition 6. [12] Q-algebra V is called:
- Abelian, or commutative Q-algebra, if X * Y = 0 (X,Y £ V);
- K-algebra, or anti-commutative Q-algebra, if X * Y = —Y * X, (X, Y £ V);
- A-algebra, or pseudo-commutative Q-algebra, if
(X * Y,Z} + (Y * Z, X} + (Z * X, Y} = 0, (X, Y,Z £ V).
We recall [13] that in the module A'(M) of an almost contact metrical manifold the structure of Q-algebra Re is naturally introduced over the ring of complex-valued smooth functions with the operation
X * Y = T(X,Y) = 1 ($)$r — ($)$2F}; X,Y £X(M) (20)
and metrics
((X, Y}} = (X, Y} + 4—1 (X, }; X,Y £ X (M). (21)
This Q-algebra is called adjoint.
Assume that M is a GK-manifold. In the C™(M) — module A'(M) of smooth vector fields of manifold M a binary operation " * " is introduced by the formula
X * Y = T(X, Y) = 1 ($)$r — ($)$2F}; X,Y £ X(M)
Theorem 2. The GK-structure has an anti-commutative adjoint Q-algebra, h.e. a K-algebra.
Proof. From Definition 2 it is easy to follow that + X,Y £ X(M),
so ($)$r = — X,Y £ X(M).
And it means that ($)$2Y = — ($)$2^ X,Y £ X(M). Then
T(X, Y) = 4— ($)$2Y} =
= — 1 — ($)$2X} = —T(Y, X); X,Y £ X(M).
H.e. the adjoint Q-algebra of GK-structures is a K-algebra.
Corollary 1. SGK-manifolds of the I kind and Kenmotsu manifolds have Abelian adjoint Q-algebra.
Proof. For SGK-manifolds of the I kind and Kenmotsu manifolds the identity is realized ([9]) $V$x= — $V$Y= 0 X,Y £ X(M), and it means that
T(X, Y) = 4{ ($)$Y — W^2x($)$2F} = 0;
X,Y £ ^(M), h.e. adjoint Q-algebra is Abelian.
Corollary 2. SGK-manifolds of the II kind have anti-commutative adjoint Q-algebra.
Proof. From (9) we have $V$X+ $V$Y($)$X; X,Y e X(M), than according to the obtained equality
X * Y = T(X, Y) = 1 {^V^x($)$r - $V$2X($)$2F} = = - 1 {$V$Y ($)$X - ($)$2X} =
= -T(Y, X) = -X * Y; X,Y e X(M),
h.e. the adjoint Q-algebra is commutative, h.e. a K-algebra.
Corollary 3. Kenmotsu manifolds have an Abelian adjoint Q-algebra. Proof. For Kenmotsu manifolds the equality is executed ([11])
T (X, Y) = 1 { $V$x ($)$r - $V$2X ($)$2 Y} = 0,X,Y e X (M), h.e. the adjoint Q-algebra is Abelian.
5. The type constancy of generalized Kenmotsu manifolds
In this section we consider a contact analog of the type constancy and examine it in detail for generalized Kenmotsu manifolds.
Definition 7. [14] The complex K-algebra Re is called the K-algebra of constant type, if 3c e C yX,Y e Re : ({X, Y)) = 0 ^ \\X * Y ||2 = c\\X ||2 \\Y ||2.
Definition 8. The GK-manifold M is called a pointlike constant type manifold, if its adjoint Q-algebra has a constant type in each point of manifold M. Function c, if it exists, is called the type constant of the GK-manifold. If c = const, than M is called a global constant type GK-manifold.
Re
operation * : A'(M) x A'(M) ^ A'(M), defined by the identity
X * Y = T(X, Y) = 4{$V$x($)$r - W$2X($)$2F};
X,Y e X(M). From (9) it follows that on the GK-manifold $V$x($)$r = -W$2X($)$2F; X,Y e X(M^^us X * Y = 2$V$x($)$r = -2 W$2X($)$2Y} X,Y e X(M). Because of (21), the condition {X, Y) = {X, ) = 0 equals the condition {{X, Y)) = 0. Thus, the following is true
Theorem 3. A GK-manifold is a manifold of pointlike constant type c then and only then, when
yX,Y e X(M) {{X,Y)) = 0 ^\\C(X,Y)\\2 = c\\X\\2 \\Y\\2 . (22)
We introduce into consideration a 4-form
C(X, Y, Z, W) = {{X * Y,Z * W)) = {{C(X, Y),C(Z, W))).
It is directly verified that it has the following properties: 1) Antilinearitv at the first pair of arguments
V-1C (X, Y, Z, W) = -C ($X, Y, Z, W) = -C (X, $Y, Z, W).
2) Linearity at the second pair of arguments
V-ÎC(X, Y, Z, W) = -C(X, Y, $Z, W) = -C(X, Y, Z, ).
3) Skew symmetry at the first and second pairs of arguments
C(X, Y, Z, W) = -C(Y, X, Z, W) = -C(X, Y, W, Z).
4) Hermicitv
C(X,Y,Z,W) = C(Z,W,X,Y),X,Y,Z,W e X(M).
Because
C(X, Y, X, Y) = {{X * Y,X * Y}} = {{C(X, Y), C(X, Y)}} = \\C(X, Y)||2 ,
GK-manifold M is of a pointlike constant type c then and only then, when
C(X,Y,X,Y) = c \\X\\2 \\Y\\2 , X,Y eX(M), {{X,Y}} = 0. (23)
We polarize this equality, replacing Y with Y + Z, where Z e X(M), {{X,Z}} = 0: C(X,Y + Z,X,Y + Z) = c \ \X\\2 \\Y + Z\\2. After distribution through its linearity and the required reduction considering (23), we get
C(X, Y, X, Z) + C(X, Z, X,Y)= c \\X\\2 ({{Y, Z}} + {{Z, Y}}). (24)
Replacing Z with $Z here, while considering the properties 1) and the non-degeneracy of $
C(X, Y, X, Z) — C(X, Z, X,Y) = c\\X\\2 (— {{Y, Z}} + {{Z, Y}}). (25)
Summing the identities (24) and (25) term by term we get:
C(X,Y,X,Z) = c\\X\\2 {{Z,Y}}. (26)
Let now Y,Z e X(M) be arbitrary vectors. Let us distribute them over the linear hull of vector X and its orthogonal complement: Y = ^^ X + Y'\ Z = X + Z' . Considering (26) and
the property 3) after the required reduction we get:
C(X,Y,X,Z) =C(X,Y',X,Z') = c\\X\\2 {{Z',Y'}} =
=o\\2a az—w x,y—wx rere=
= c{{{Z,Y }}\\X\2 — {{Z,X }}{{X,Y}}}.
So,
C(X,Y,X,Z) = c{{{Z,Y}} \\X\2 — {{Z,X}} {{X,Y}}}. (27)
Let us replace Z with W in the obtained equality, then
C(X,Y,X,W) = c{{{W,Y}} \\X\2 — {{W,X}} {{X,Y}}}.
In the last identity we replace X with X + Z and after removal through linearity and after the required reduction while considering (27) we get:
C(X, Y, Z, W) = c{{{W, Y}} {{Z, X}} — {{W, X}} {{Z, Y}}}. (28)
Inverse, it is obvious that because (28), (23) is fulfilled, thus, M is a GK-manifold of a pointlike constant type c.
Thus the following theorem is proved.
Theorem 4. A GK-manifold is a manifold of pointlike constant type c then and only then, when the following is realized
C(X, Y, Z, W) = {{C(X, Y),C(Z, W))) c{{{W, Y)) {{Z, X)) --{{W,X )){{Z,Y))}.
We introduce the following theorem giving the structural tensor properties. Theorem 5. GK-structure structural tensors have the following properties:
1) $ o C(X, Y) = -C($X, Y) = -C(X, $Y); 2) $ o F = -F o $; 3) {{C(X,Y),Z)) + {{Y,C(X,Z))) = 0; (29)
4) F(0 = 0;
5) ^ o F = 0.
Proof. 1) After covariant differentiation of the equality $2 = -id + ® we get VY($)$X + $ o V$Y($)X = £Vy(-q)X + -q(X)VYIn the to equality we change X ^ $X, and the received identity will be influenced by operator $2, then we get $oVY($)$X = $2oVY($)$2X. In the received identity we change Y ^ $Y, then
$ o V$Y($)$X = $2 o V$Y($)$2X; VX,Y e X(M). (30)
In the received identity we change X ^ $X, then we get $oV$Y($)$2X = -$2oV$Y($)$X; yx, Y e X(M). Considering the last identity, from (19) we have
$ o c(X, Y) = -1$2 o V$Y($)$x = -1$ o V$Y($)$2X = -C($X, Y)
and
C($X, $) = -1 $ o V$y($)$2X = 1 $ o V$2X($)$F = = 2 $2 o V$2X ($)$2F = - 2 $2 o V$2Y ($)$2 X = = -1 $ o V$2Y($)$X = c(X, $Y).
2) Prom the analytic expression of the second structural tensor and from the expression F(X) = -$2 o V$x($)£ - $2X = -$ o V$x($)£ - $2^; yX,Y e X(M), it follows that $ o F = -F o $.
3) Considering (18) and (21) we have
{{C(X, Y), Z)) = {C(X, Y), Z) + V=T {C(X, Y), $Z) = = (-1 $ o V$Y($)$x, Z) + y/-[ (-1 $ o V$Y($)$x, $Z) =
= -2 {$x, V$y($)$z) + 2y- ($x, V$y($)$2^> = = 2 {X, $ o V$Y($)$z) - 1 y=T (X, $ o V$Y($)$2^> = = 2 {X, $ o V$Y($)$z) - 1 y-T (X, $2 o V$Y($)$z) = = - {X, C(Y, Z)) - y-T {X, $C(Y, Z)) = -{{X,C(Y,Z))).
4) Because $(£) = 0, we have F(£) = -$ o ($)£ - $2{ = 0.
5) Because the almost contact metrical manifold has equalities o $ = 0 and ~q(Vx£) = 0, then rj(F(X)) = -rj(V x0 - rj($2X) = 0, h.e. q o F = 0.
Let us find the representation of the equality (28) of the adjoint G-structure. We fix the point p £ M, orthonormal frame r = (p, el,..., en) of space TP(M), that is considered as a C-module, and the corresponding A-frame
r = (p,€0,€l, ...,€n,€l, ...,€n),
where ea = \f2av(ea), ea = V2ap(ea), eo = Putting into (28) X = ea, Y = eb, Z = ec, W = ed, we get an equivalent (in point p) identity
C(ea,eb,ec,ed) = ((С (ea,eb ),C (ec,ed))) =
= c{((ed, еь)) ((ec, ea)) - ((ed, ea)) ((ec, eb))}. (31)
Because
((ea, eb)) = (<JZa, oeb) + V-l (aea, aeb) = 2 (aea, aeb) = (ea, eb^ = ^, which also comes from the orthonormal nature of the frame r. Because Сabc = ±C(e-b, ea)a, then
((C(ea,eb),C(ec,ed))) = (C(ea,eb),C(ec,ed)) + V-l (C(ea,eb),C(ec,ed)) = = 2 (aC(ea, еь),аС(ec, ed)) = 2 (C(aea, аеь),С(aec, aed)) = = l (C(ea, еь), С(ec, €d)) = 2 (Chabeh, Cg^e") =
_ ^sfhabrf /, rg\ _ r>f~ihabri _ ^riabhri
= 2Ь bgcd (eh, еУ ) = 2b bgcd = 2Ь bhcd.
Then relations (31) can be formulated as CabhChcd = |§ab, where = ¿a^d ~ ^c^a '1S a Kronecker delta of the second order. Through this we prove the following theorem.
тн eorem 6. Let M be a GK-manifold. Then the following statements are equivalent:
1) M is a GK-manifold of a pointlike constant type c.
2) The first structural tensor of a GK-manifold satisfies the identity
((C(X,Y),C(Z,W))) = c{((W,Y)) ((Z,X)) - ((W,X)) ((Z,Y))}.
3) On the adjoint G-structure space the following relation is correct
CabhChcd = ¿Sc dab. (32)
Let us differentiate externally the following equality (32)
dCabhChcd + CabhdChcd = 2dc5ab-Considering the structural equations of GK-manifolds we have
(-Cgbhe'a - Cagheg - Cabgehg + Cabhgug - 25^Fbh]wg --Cabhu)Chcd + Cabh(CgCddl + ChgdOgc + Chcg09g + ChcdgUj" --26ghFcd]Ug - ChcdU) = 2Sabdc.
Opening the brackets and collecting similar terms considering (32), (12l0), (12l4), (11), we have
(CabhChcgd - 3FabCgCd)ujg - (CahgbChcd + IcabgFcd)ug - 4c5ab =
= 2Sab(cg ug + cg Wg + cou).
From here we have
l) fe = CabhChcgd - 3FabCgcd; (33)
2) c" = ~cahgbChcd - 3FcdCabg; 3) Co = -2c.
Contracting equalities (33i) and (ЗЗ2) at first by indices a and c, and then bv indices b and d, considering the equalities FadCdbc = FadCdbc = 0 Mid CabcgCgdh = CabcgCgdh = 0, we get n(n — 1)Cg = n(n — 1)cg = 0, h.e. either dimM = 3, or cg = c9 = 0.
Thus dc = — 2c. Differentiating the last equality externally we get cdw = 0, which, considering (1(h) will be written as: c(Fabua Л wb + Fabua Л шь) = 0, h.e. cFabua Л шь + cFabua Л шь = 0.
Therefore either с = 0, or Fab = Fab = 0.
Thus the following theorem is proved.
Theorem 7. AGK-manifold of a constant non-zero type is a SGK-manifold of the II kind.
Assume that M is a SGK-manifold of the II kind then we will perform a complete convolution (23): ^2abc\Cabc\2 = CabcCabc = 2cn(n — 1), where и is a complex dimensionality of the contact distribution L. This implies that с > 0 с £ R, at that с = 0 then and only then, when Cabc = 0, h.e. M is a Kenmotsu manifold. Thus it's proved that
Theorem 8. The class of SGK-manifolds of the II kind of the zero constant type coincides with the class of Kenmotsu manifolds.
We have to investigate SGK-manifolds of the II kind of the non-zero constant type. According to [4] SGK-manifolds of the II kind of the non-zero constant type are concircularlv transformed into most precise cosvmplectic manifolds which, in turn, are locally equivalent to the product of own (h.e. non-Keller) almost Keller eigenmanifold and the real straight line. Because the class of almost Keller manifolds of the non-zero constant type coincides with the class of six-dimensional almost Keller eigenmanifolds ([3], [14]), we can formulate the following theorem.
Theorem 9. The class of SGK-manifolds of the II kind of the zero constant type coincides with the class of Kenmotsu manifolds. The class of SGK-manifolds of the II kind of the non-zero constant type is concircularly transformed into the almost contact metrical manifolds locally equivalent to the product of the six-dimensional N К-eigenmanifold and the real straight line.
Theorems 8 and 9 can be combined into the following Fundamental theorem.
Theorem 10. Fundamental theorem. The class of GK-manifolds of the zero constant type coincides with the class of Kenmotsu manifolds. The class of GK-manifolds of the non-zero constant type is concircularly transformed into the almost contact metrical manifolds locally equivalent to the product of the six-dimensional N К-eigenmanifold and the real straight line.
6. Conclusions
In this work fully research generalized Kenmotsu manifolds constancy of type. The local characteristic of these manifolds is obtained. The main theorem gives a complete solution to the assigned task.
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Получено 6.03.2019 г.
Принято в печать 12.07.2019 г.
